Critical Properties of the Measurement-Induced Transition in Random Quantum Circuits

A transition was observed in random quantum circuits with local projective measurements where the entanglement entropy (EE) goes from volume law to area law scaling as the measurement rate, p, is increased above a critical value, p_c. Attempts to extract the critical properties through finite size scaling of the EE has proven difficult due to the logarithmic divergence at criticality. We study the tripartite mutual information (TMI) as an alternative diagnostic of the transition that is finite at criticality while maintaining a volume law for p<p_c and vanishing for p>p_c. Our intuition for the Haar random circuit is guided by results on stabilizer circuits which can be simulated at much larger sizes. We find that the TMI has weaker finite size effects when compared to other quantities. Our numerics of the Haar random circuit suggests p_c≈0.17 and the critical exponent ν≈1.3 for Renyi indices n≥1, which are smaller than previously reported results. We also find a strong Renyi index dependence of the coefficient of the logarithmic divergence α(n)=1.0(1)/n+0.7(1).